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Copyright by Timothy Alexander Cousins 2016 Copyright by Timothy Alexander Cousins 2016 The Thesis Committee for Timothy Alexander Cousins Certifies that this is the approved version of the following thesis: Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media APPROVED BY SUPERVISING COMMITTEE: Supervisor: Hugh Daigle Maša Prodanović Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media by Timothy Alexander Cousins, B. S. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2016 Dedication I would like to dedicate this to my parents, Michael and Joanne Cousins. Acknowledgements I would like to thank the continuous support of Professor Hugh Daigle over these last two years in guiding throughout my degree and research. I would also like to thank Behzad Ghanbarian for being a great mentor and guide throughout the entire research process, and for constantly giving me invaluable insight and advice, both for the research and for life in general. I would also like to thank my parents for the consistent support throughout my entire life. I would also like to acknowledge Edmund Perfect (Department of Earth and Planetary, University of Tennessee) and Jung-Woo Kim (Radioactive Waste Disposal Research Division, Korea Atomic Energy Research Institute) for providing Lacunarity MATLAB code used in this study. v Abstract Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media Timothy Alexander Cousins, M.S.E. The University of Texas at Austin, 2016 Supervisor: Hugh Daigle Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as in a network of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. A fractal medium is one whose pore space and solid matrix can be characterized by statistical self-similarity and described by a fractal dimension Dm. Specifically, in a random mass fractal, each iteration of construction of the medium is composed of identical-size particles and pores of different sizes that are distributed vi randomly within (Hunt et al. 2014). This thesis contains the research into the effect of rough pore-solid interface on single-phase flow and permeability in fractal porous media. Using fractal geometry, randomly generated three-dimensional Menger sponges were created to model porous media with a range of mass fractal dimensionalities Dm between 2.579 and 2.893. This dimensionality characterizes both the solid matrix and the pore space of the media. The pore-solid interface of the media is subsequently roughened using the Weierstrass-Mandelbrot approach and controlled primarily by the surface fractal dimension Ds and root-mean-square of roughness height σ. The permeability was calculated for all the roughened media using the lattice- Boltzmann method using D3Q19 geometry and Bhatnagar-Gross-Krook (BGK) collision model. The LBM simulations calculated the single-phase permeability based on Darcy’s Law. Results indicate that permeability decreases sharply with increasing Ds from 1 to 1.1 regardless of Dm value, and remains relatively constant as Ds increases from 1.1 to 1.6. Furthermore, while creating the media, a lower bound for the percolation threshold appeared to be around 29.8% for randomized Menger sponges. When fitted to the percolation model presented in Larson et al. (1981) with an upper limit of 0.36 from Kim et al. (2011), the parameters from a least squares fit point to a critical porosity ϕc of 30% and a percolation exponent t between 3.1 and 3.3. Future research should investigate the effect of the percolation threshold for these simulated porous media and the effect surface roughness would have on this threshold. Finally, future research should expand into two-phase flow and investigate the effects of vii surface roughness on relative permeability and capillary pressure in simulated fractal porous media. viii Table of Contents List of Tables ......................................................................................................... xi List of Figures ....................................................................................................... xii Chapter 1: Introduction ........................................................................................... 1 Chapter 2: Background ........................................................................................... 7 2.1 Permeability .......................................................................................... 7 2.2 Fractal Geometry .................................................................................. 9 2.2.1 Fractal Porous Media Forms ...................................................... 13 Sierpinski Carpet ........................................................................ 15 Menger Sponge .......................................................................... 19 2.2.2 Pore-Surface Roughness ............................................................ 22 2.2.3 Lacunarity .................................................................................. 26 2.3 Lattice-Boltzmann Simulation ............................................................ 28 Chapter 3: Methods ............................................................................................... 34 3.1 Creating the Fractal Porous Media ..................................................... 34 3.2 Roughening the Pore-Solid Interface .................................................. 37 3.3 Lattice-Boltzmann Simulations in Palabos ......................................... 45 Chapter 4: Results and Discussions ...................................................................... 47 4.1 LBM Simulation Results ..................................................................... 47 4.2 Discussion ........................................................................................... 52 4.2.1 Permeability versus Surface Roughness Fractal Dimension ..... 52 4.2.2 Percolation Threshold ................................................................ 55 Chapter 5: Conclusions ......................................................................................... 65 Appendix ............................................................................................................... 68 A1. Matlab Code for Creating Media ........................................................ 68 A2. Matlab Code for Roughening Pores .................................................... 70 ix A3. Matlab Code to convert to media to .dat file (Source: Palabos) ......... 79 A4. Master Script to Create Menger sponge .............................................. 85 Glossary ................................................................................................................ 86 Bibliography ......................................................................................................... 87 x List of Tables Table 3.1: Properties of 3D randomized Menger sponges ................................. 35 Table 4.1: LBM permeability results for x-direction ......................................... 47 Table 4.2: LBM permeability results for y-direction ......................................... 47 Table 4.3: LBM permeability results for z-direction ......................................... 47 Table 4.4: Normalized permeability for x-direction .......................................... 50 Table 4.5: Normalized permeability for y-direction .......................................... 51 Table 4.6: Normalized permeability for z-direction .......................................... 51 Table 4.7: Ds vs Permeability for Dm=2.7712 for different σ/d ......................... 54 Table 4.8: Optimized parameters of Equation (4.1) for x direction ................... 57 Table 4.9: Optimized parameters of Equation (4.1) for y direction ................... 57 Table 4.10: Optimized parameters of Equation (4.1) for z direction .................. 57 Table 4.11: Optimized parameters of Equation (4.1) with ϕc = 0.36. ................. 58 Table 4.12: Kozeny-Carman Permeability results .............................................. 60 xi List of Figures Figure 2.1: Visualization of self-similarity of a Koch Snowflake (figure courtesy of B. Ghanbarian). .................................................................................. 9 Figure 2.2: Measuring coastline of Great Britain with ε = 200, 100 & 50 km (Source: http://intothefractalvoid.blogspot.com/2012/11/infinity_24.html). .. 10 Figure 2.3: Visualization of Euclidean versus Fractal Dimensionality (figure courtesy of B. Ghanbarian). ............................................................. 11 Figure 2.4: Qualitative description of statistical self-affinity for a surface profile (Chen et al. 2009). ............................................................................ 13 Figure 2.5: Perfectly self-similar Sierpinski Carpet with D = 1.893 as a (a) mass fractal and a (b) pore fractal with solid matrix in black and pore space in white (Hunt et al. 2014). .................................................................
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